The set of values of ' x ' for which function f(x)=x ln x-x+1 takes positive value only, is

Question

The set of values of ' x ' for which function f(x)=x ln x-x+1 takes positive value only, is (A) (1, ∞) (B) ((1/ e ), ∞) (C) [e, ∞) (D) (0,1) ∪(1, ∞)

Tardigrade Admin · Accepted Answer

f ( x )= x ln x - x +1 ; f (1)=0 f prime( x )=1+ ln x -1= ln x f prime( x )=<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/476df674739b7a9484de5eb287b58c92-.png" /> ⇒ f ( x ) text is decreasing for (0,1) text and increasing for (1, ∞) f ( x ) min = f (1)=0 f ( x ) ≥ 0 ∀ x ∈(0, ∞)

Q. The set of values of ' $x$ ' for which function $f (x) = x ln x - x + 1$ takes positive value only, is

$(1, \infty)$

$(\frac{1}{e}, \infty)$

$[e, \infty)$

$(0, 1) \cup (1, \infty)$

$f (x) = x ln x - x + 1; f (1) = 0$
$f^{'} (x) = 1 + ln x - 1 = ln x$
$f^{'} (x) =$
$\Rightarrow f (x) is decreasing for (0, 1) and increasing for (1, \infty)$
$f (x)_{m i n} = f (1) = 0$
$f (x) \geq 0\forall x \in (0, \infty)$

Q. The set of values of ' x ' for which function f(x)=xlnx−x+1 takes positive value only, is

(1,∞)

(e1​,∞)

[e,∞)

(0,1)∪(1,∞)